Estimating Welfare in Insurance Markets
using variation in prices
Einav, Finkelstein, and Cullen (QJE, 2010)
February 28, 2024
Overview
- Objective: Quantify the welfare loss in insurance markets with (adverse or advantageous) selection due to inefficient pricing.
- Key Findings:
- Evidence of adverse selection in employer-provided health insurance.
- Quantitative welfare loss of inefficient pricing is small.
- Methodology:
- Use of price variation to identify and estimate demand and insurers’ costs.
Overview
- Contribution:
- Show how to use standard consumer and producer theory to empirically estimate welfare losses in markets with selection.
- According to authors: Fewer assumptions about consumers’ preferences, straightforward implementation, and general
Introduction
- Complementary approach to empirical welfare analysis requiring fewer assumptions
- Revealed preference, demand and cost curves are sufficient statistics for welfare analysis
- Fewer assumptions at the cost of limiting welfare analysis to only observed (existing) contracts
- Distortion of offered contracts set not allowed
Introduction
- In Graphical analysis, the authors show that the slope of the cost curve provides a direct test of selection
- Constant (horizontal) marginal cost curve, no selection; negative, adverse selection; positive, advantageous selection
Introduction
- Employer-provided health insurance
- Individual-level data from Alcoa, Inc., large multinational aluminum producer
- Health insurance options, choices, and medical insurance claims
- Price variation used in identification arises due to Alcoa’s organizational structure, (orthogonal to employee characteristics)
- Authors detect adverse selection and estimate the welfare loss to be small (<$10 per employee, 3% of efficient total surplus)
Theoretical Framework
- Individuals choose from two insurance contracts, \(H\) high (insurance) and \(L\) low coverage (no insurance).
- Characteristics of insurance contracts are taken as given, but price is determined endogenously
- Population defined by a distribution \(G(\zeta)\), \(\zeta\) vector of consumer characteristics
- Relative price of contract H by \(p\)
Theoretical Framework
- \(v^H(\zeta_i,p)\) and \(v^L(\zeta_i)\) denote the consumer \(i's\) utility from buying contracts \(H\) and \(L\), respectively
- \(v^H(\zeta_i,p)\) strictly decreasing in \(p\) and \(v^H(\zeta_i,p=0)>v^H(\zeta_i)\)
- Expected monetary cost of providing insurance to individual \(i\), \(c(\zeta_i)\)
Demand for Insurance
- Each individual makes a discrete choice, buying insurance or not
- Demand is only a function of price (abstracting from coverage level)
- Firms cannot offer different prices to different individuals (empirical application: exogenous price variation)
- Individuals buy at the lowest price available, demand can be sufficiently characterized as a function of the lowest price \(p\)
- Individual \(i\) buys insurance if and only if \(v^H(\zeta_i,p)>v^L(\zeta_i)\)
- Willingness to pay \(\pi(\zeta_i)\equiv\max\{p:v^H(\zeta_i,p)>v^L(\zeta_i) \}\)
Demand for Insurance
- Aggregate demand is \[
D(p)=\int \mathbb{1}(\pi(\zeta)\ge p)dG(\zeta)=\text{Pr}(\pi(\zeta)\ge p)
\qquad(1)\]
Supply
- \(N\ge2\) insurance providers set prices à la Bertrand (perfect competition) Under perfect competition, any inefficiency is attributed to selection
- The cost of providing contract \(H\) to individual \(i\) are the insurable cost \(c(\zeta_i)\) No other admin or production costs
Supply
The average (expected) cost curve \[
\text{AC}(p)=\frac{1}{D(p)}\int c(\zeta)\mathbb{1}(\pi(\zeta)\ge p)dG(\zeta)=\mathbb{E}[c(\zeta)| \pi(\zeta)\ge p]
\qquad(2)\]
The marginal (expected) cost curve is \[
\text{MC}(p)=\mathbb{E}[c(\zeta)|\pi(\zeta)=p]
\qquad(3)\]
Equilibrium
- It is profitable to provide insurance
- Single crossing between MC\((p)\) and the demand curve
- The equilibrium \[
p^*=min\{p:p=\text{AC}(p)\}
\]
Measuring Welfare
- Authors measure consumer surplus by the certainty equivalent
- The certainty equivalent of an uncertain outcome is the amount that would make an individual indifferent between obtaining this amount for sure and obtaining the uncertain outcome
- Income effects are ignored (equivalent to assuming CARA utility function)
- \(e^H(\zeta_i)\) and \(e^L(\zeta_i)\) denote the certainty equivalent for consumer \(i\) of an allocation of contract \(H\) and \(L\), respectively
- The willingness to pay \(\pi(\zeta_i)=e^H(\zeta_i)-e^L(\zeta_i)>0\)
Measuring Welfare
- Consumer welfare \[
\text{CS}=\int[(e^H(\zeta)-p)\mathbb{1}(\pi(\zeta)\ge p)+e^L(\zeta)\mathbb{1}(\pi(\zeta)<p)]dG(\zeta)
\]
- Producer welfare \[
\text{PS}=\int(p-c(\zeta))\mathbb{1}(\pi(\zeta)\ge p)dG(\zeta)
\]
Measuring Welfare
- Total welfare \[
\text{TS=CS+PS}=\int[(e^H(\zeta)-c(\zeta))\mathbb{1}(\pi(\zeta)\ge p)+e^L(\zeta)\mathbb{1}(\pi(\zeta)<p)]dG(\zeta)
\]
Graphical Representation
- Adverse selection: Individuals with the highest willingness to pay for insurance are those who, on average, have the highest expected costs
- Downward-sloping MC: MC is decreasing in quantity
- As the price falls, the marginal individuals who select contract \(H\) have a lower expected cost than inframarginal individuals, leading to lower average costs
- Uniform pricing (firms cannot discriminate individuals based on their privately known marginal costs) implies average-cost pricing in equilibrium
- AC > MC leads to underinsurance in adverse selection
![]()
Adverse selection implies the marginal cost curve is downward- sloping, people with the highest willingness to pay also have the highest expected cost to the insurer. Competitive equilibrium is given by point C (demand =average cost), whereas the efficient allocation is E (demand = marginal cost). The (shaded) triangle CDE represents the welfare cost from underinsurance.
Advantageous Selection
- Individuals with the highest willingness to pay insurance have, on average, the lowest expected costs.
- MC and AC curves are upward-sloping
- Leading to over-insurance
- Intuitively, insurance providers have an additional incentive to reduce price, as the inframarginal customers whom they acquire as a result are relatively good risks.
![]()
Advantageous selection implies the marginal cost curve is upward- sloping, people with the highest willingness to pay also have the lowest expected cost to the insurer. Competitive equilibrium is given by point C (demand =average cost), whereas the efficient allocation is E (demand = marginal cost). The (shaded) triangle CDE represents the welfare cost from overinsurance.
Sufficient Statistics
- Graphical analyses illustrate that the demand and cost curves are sufficient statistics for welfare analysis of equilibrium and nonequilibrium pricing of existing contracts
- Different underlying primitives have the same welfare implications if they generate the same demand and cost curves
- Primitives: preferences and private information summarized by \(\zeta\)
Sufficient Statistics
- This suggests an empirical approach to remain agnostic about the underlying primitives: estimating demand and cost curves
- Requiring: Revealed preferences choices from individuals can be used for welfare analysis
- Limits counterfactuals to only price variation for the set of observed insurance contracts
- Examples: Mandatory insurance coverage, taxes and subsidies for insurance, regulation eliminating an existing contract, etc.
Estimation
- The framework requires data allowing the estimation of the demand curve \(D(p)\) and the average cost curve \(AC(p)\)
- Marginal cost curve can be derived by \[
\begin{aligned}
MC(p)
&=\frac{\partial TC(p)}{\partial D(p)}=\frac{\partial(AC(p)\cdot D(p))}{\partial D(p)}\\
&=\left(\frac{\partial D(p)}{\partial p}\right)^{-1}\frac{\partial(AC(p)\cdot D(p))}{\partial p}
\end{aligned}
\]
Identification
- To identify demand, price variation exogenous to unobservable demand characteristics is needed
- Authors claim they do not need an additional source of variation to estimate the average cost curve AC(p) because
- Expected costs are likely to affect demand, any exogenous price variation to demand is also exogenous to insurable cost
Employer-Provided Health Insurance
- Individual level data from 2004 on the U.S.-based employees and their dependants at Alcoa, Inc.
- In 2004, Alcoa had ~45,000 employees in the U.S. working in 300 different job sites in 39 different states.
- Alcoa introduced a new set of health insurance options for virtually all its salaried employees and about half of its hourly employees
- Data includes the menu of health options available, employee premium of each option, employee’s coverage choice, and detailed claim-level information on all the employee medical expenditures including covered dependants
- Data also includes demographic information, such as age, race, gender, annual earnings, and job tenure
Subsample
- Analysis restricted to a subsample of employees for whom the pricing variation is cleaner
- 3,779 salaried employees who chose one of the two modal health insurance choices: a higher and a lower level of PPO coverage
Price Variation
- In 2004, as part of the new benefit design, company headquarters offered a set of seven different possible pricing menus for employee benefits.
- The coverage options are the same across all the menus, but the prices (employee premiums) associated with these options vary
- The object of interest is the incremental (annual) premium for contract \(H\) relative to \(L\), \(p=p_H-p_L\)
- There were six different values of \(p\) in 2004, from $384 to $659
Price Variation by Business Unit
- The president of each business unit determined the price menu for the employees of their respective business unit
- Alcoa had forty business units. Each unit has essentially complete independence to run its business
- Business units are typically organized by functionality and are independent of geography (there are often multiple business units in the same state)
Price Variation by Business Unit
- The median business unit has about 500 employees
- The business unit president may choose different price menus for employees within his unit based on their location and employment type (salaried/hourly, hourly by union)
- As a result, employees doing the same job in the same location may face different prices for their health insurance benefits due to their business affiliations
Price Variation
Price Variation Analysis
- Authors claim that they could not reject the null that all the coefficients are jointly uncorrelated; F-test p-value=0.14
- Business unit presidents were opposed to charging employees much for health insurance coverage
- After 2004, Alcoa no longer gave business unit presidents a choice on benefit prices and adopted a uniform pricing structure
- A similar analysis for hourly workers reveals statistically significant differences across employees who face different prices
Price Variation Analysis
Empirical Strategy
- \(p_i=p_i^H-p_i^L\), relative price employee \(i\) faces
- \(D_i\) is equal to 1 if employee \(i\) chooses contract \(H\), and O if \(L\)
- \(m_i\) total medical expenditure of employee \(i\) and covered family members
- \(c_i=c(m_i;H)-c(m_i;L)\) is the incremental cost to the insurer from providing contract H relative to L, holding \(m_i\) fixed.
- \(c(m_i;H)\) and \(c(m_i;L)\) denote the cost to the insurance company from medical expenditures under H and L
- \(c(m_i;H)\) is observed; \(c(m_I;L)\) is computed counterfactually using the claims data and the plan L rules
Estimating Equations
\[
\begin{aligned}
D_i&=\alpha+\beta p_i + \epsilon_i \\
c_i&=\gamma+\delta p_i + u_i
\end{aligned}
\qquad(4)\]
MC and Equilibrium P and Q
Having estimated the demand and AC, we can find MC,
\[
MC(p)=\frac{\alpha\delta}{\beta}+\gamma+2\delta p
\]
With MC, we can find the equilibrium and the efficient price and quantity; Equilibrium AC(p)=p, Efficient MC(p)=p \[
\begin{aligned}
P_{eq} &=\gamma/(1-\delta)\\
Q_{eq} &=\alpha + \beta(\gamma/(1-\delta))\\
\end{aligned}
\]
Efficient P and Q
Efficient MC(p)=p \[
\begin{aligned}
P_{eff} &=\frac{1}{(1-2\delta)(\frac{\alpha\delta}{\beta}+\gamma)}\\
Q_{eff} &=\alpha+\frac{1}{(1-2\delta)(\alpha\delta+\beta\gamma)}
\end{aligned}
\]
Welfare Loss \[
\Delta_{CDE}=-\frac{\delta^2}{2(1-2\delta)\beta}\left(\alpha+\frac{\beta\gamma}{1-\delta}\right)^2
\]
Results
Results
Results
- Downward sloping demand \(\beta=-0.0007\)
- a $100 increase in price reduces the probablity of getting \(H\) by 11%
- Cost curve slope indicates the existence of adverse selection \(\delta=0.155\)
- The average cost of individuals who purchased contract \(H\) is higher when the price is higher
- The average cost curve is therefore downward-sloping in quantity
Results
- Welfare cost of adverse selection $9.55 per employee per year
- Adverse selection increases price by almost $200 and reduces the share of H by 14 p.p.